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Theorem cnextf 20750
Description: Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
Hypotheses
Ref Expression
cnextf.1  |-  C  = 
U. J
cnextf.2  |-  B  = 
U. K
cnextf.3  |-  ( ph  ->  J  e.  Top )
cnextf.4  |-  ( ph  ->  K  e.  Haus )
cnextf.5  |-  ( ph  ->  F : A --> B )
cnextf.a  |-  ( ph  ->  A  C_  C )
cnextf.6  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  C )
cnextf.7  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
Assertion
Ref Expression
cnextf  |-  ( ph  ->  ( ( JCnExt K
) `  F ) : C --> B )
Distinct variable groups:    x, A    x, B    x, C    x, F    x, J    x, K    ph, x

Proof of Theorem cnextf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cnextf.3 . . . 4  |-  ( ph  ->  J  e.  Top )
2 cnextf.4 . . . 4  |-  ( ph  ->  K  e.  Haus )
3 cnextf.5 . . . 4  |-  ( ph  ->  F : A --> B )
4 cnextf.a . . . 4  |-  ( ph  ->  A  C_  C )
5 cnextf.1 . . . . 5  |-  C  = 
U. J
6 cnextf.2 . . . . 5  |-  B  = 
U. K
75, 6cnextfun 20748 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Haus )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  Fun  ( ( JCnExt K
) `  F )
)
81, 2, 3, 4, 7syl22anc 1231 . . 3  |-  ( ph  ->  Fun  ( ( JCnExt
K ) `  F
) )
9 simpl 455 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  ph )
10 cnextf.6 . . . . . . . . 9  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  C )
1110eleq2d 2472 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  C ) )
1211biimpar 483 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  ( ( cls `  J
) `  A )
)
13 cnextf.7 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
14 n0 3747 . . . . . . . 8  |-  ( ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/)  <->  E. y 
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F ) )
1513, 14sylib 196 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  E. y 
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F ) )
16 haustop 20017 . . . . . . . . . . . . . 14  |-  ( K  e.  Haus  ->  K  e. 
Top )
172, 16syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  Top )
185, 6cnextfval 20746 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  (
( JCnExt K ) `
 F )  = 
U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
191, 17, 3, 4, 18syl22anc 1231 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( JCnExt K
) `  F )  =  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2019eleq2d 2472 . . . . . . . . . . 11  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
)  <->  <. x ,  y
>.  e.  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
21 opeliunxp 4994 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2220, 21syl6bb 261 . . . . . . . . . 10  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
)  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
2322exbidv 1735 . . . . . . . . 9  |-  ( ph  ->  ( E. y <.
x ,  y >.  e.  ( ( JCnExt K
) `  F )  <->  E. y ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
24 19.42v 1799 . . . . . . . . 9  |-  ( E. y ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2523, 24syl6bb 261 . . . . . . . 8  |-  ( ph  ->  ( E. y <.
x ,  y >.  e.  ( ( JCnExt K
) `  F )  <->  ( x  e.  ( ( cls `  J ) `
 A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
2625biimpar 483 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( cls `  J
) `  A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )  ->  E. y <. x ,  y >.  e.  ( ( JCnExt K ) `
 F ) )
279, 12, 15, 26syl12anc 1228 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)
2825simprbda 621 . . . . . . 7  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  x  e.  ( ( cls `  J
) `  A )
)
2911adantr 463 . . . . . . 7  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  C ) )
3028, 29mpbid 210 . . . . . 6  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  x  e.  C )
3127, 30impbida 833 . . . . 5  |-  ( ph  ->  ( x  e.  C  <->  E. y <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
) ) )
3231abbi2dv 2539 . . . 4  |-  ( ph  ->  C  =  { x  |  E. y <. x ,  y >.  e.  ( ( JCnExt K ) `
 F ) } )
33 dfdm3 5132 . . . 4  |-  dom  (
( JCnExt K ) `
 F )  =  { x  |  E. y <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
) }
3432, 33syl6reqr 2462 . . 3  |-  ( ph  ->  dom  ( ( JCnExt
K ) `  F
)  =  C )
35 df-fn 5528 . . 3  |-  ( ( ( JCnExt K ) `
 F )  Fn  C  <->  ( Fun  (
( JCnExt K ) `
 F )  /\  dom  ( ( JCnExt K
) `  F )  =  C ) )
368, 34, 35sylanbrc 662 . 2  |-  ( ph  ->  ( ( JCnExt K
) `  F )  Fn  C )
3719rneqd 5172 . . 3  |-  ( ph  ->  ran  ( ( JCnExt
K ) `  F
)  =  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
38 rniun 5355 . . . 4  |-  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  = 
U_ x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
39 vex 3061 . . . . . . . . 9  |-  x  e. 
_V
4039snnz 4089 . . . . . . . 8  |-  { x }  =/=  (/)
41 rnxp 5376 . . . . . . . 8  |-  ( { x }  =/=  (/)  ->  ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
4240, 41ax-mp 5 . . . . . . 7  |-  ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )
4311biimpa 482 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  x  e.  C )
446toptopon 19618 . . . . . . . . . . 11  |-  ( K  e.  Top  <->  K  e.  (TopOn `  B ) )
4517, 44sylib 196 . . . . . . . . . 10  |-  ( ph  ->  K  e.  (TopOn `  B ) )
4645adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  K  e.  (TopOn `  B )
)
475toptopon 19618 . . . . . . . . . . . 12  |-  ( J  e.  Top  <->  J  e.  (TopOn `  C ) )
481, 47sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  C ) )
4948adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  J  e.  (TopOn `  C )
)
504adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  A  C_  C )
51 simpr 459 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  C )
52 trnei 20577 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  C )  /\  A  C_  C  /\  x  e.  C )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
5352biimpa 482 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  C )  /\  A  C_  C  /\  x  e.  C )  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
5449, 50, 51, 12, 53syl31anc 1233 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
553adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  F : A --> B )
56 flfelbas 20679 . . . . . . . . . . 11  |-  ( ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  /\  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  -> 
y  e.  B )
5756ex 432 . . . . . . . . . 10  |-  ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  ->  (
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F )  ->  y  e.  B ) )
5857ssrdv 3447 . . . . . . . . 9  |-  ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  C_  B
)
5946, 54, 55, 58syl3anc 1230 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  C_  B
)
6043, 59syldan 468 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ( ( K  fLimf  ( ( ( nei `  J ) `
 { x }
)t 
A ) ) `  F )  C_  B
)
6142, 60syl5eqss 3485 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ran  ( { x }  X.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6261ralrimiva 2817 . . . . 5  |-  ( ph  ->  A. x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
63 iunss 4311 . . . . 5  |-  ( U_ x  e.  ( ( cls `  J ) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B 
<-> 
A. x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6462, 63sylibr 212 . . . 4  |-  ( ph  ->  U_ x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6538, 64syl5eqss 3485 . . 3  |-  ( ph  ->  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6637, 65eqsstrd 3475 . 2  |-  ( ph  ->  ran  ( ( JCnExt
K ) `  F
)  C_  B )
67 df-f 5529 . 2  |-  ( ( ( JCnExt K ) `
 F ) : C --> B  <->  ( (
( JCnExt K ) `
 F )  Fn  C  /\  ran  (
( JCnExt K ) `
 F )  C_  B ) )
6836, 66, 67sylanbrc 662 1  |-  ( ph  ->  ( ( JCnExt K
) `  F ) : C --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405   E.wex 1633    e. wcel 1842   {cab 2387    =/= wne 2598   A.wral 2753    C_ wss 3413   (/)c0 3737   {csn 3971   <.cop 3977   U.cuni 4190   U_ciun 4270    X. cxp 4940   dom cdm 4942   ran crn 4943   Fun wfun 5519    Fn wfn 5520   -->wf 5521   ` cfv 5525  (class class class)co 6234   ↾t crest 14927   Topctop 19578  TopOnctopon 19579   clsccl 19703   neicnei 19783   Hauscha 19994   Filcfil 20530    fLimf cflf 20620  CnExtccnext 20743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-map 7379  df-pm 7380  df-rest 14929  df-fbas 18628  df-fg 18629  df-top 19583  df-topon 19586  df-cld 19704  df-ntr 19705  df-cls 19706  df-nei 19784  df-haus 20001  df-fil 20531  df-fm 20623  df-flim 20624  df-flf 20625  df-cnext 20744
This theorem is referenced by:  cnextcn  20751  cnextfres  20752
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